근방

일반위상수학에서 근방(近傍, 영어: neighborhood)은 어떤 점의 주위를 포함하는 집합이다. 어떤 점에 대한 근방이라는 것은 그 점을 포함하는 집합이 있어서 그 점에서 집합을 벗어나지 않은 채 어느 정도 '움직일' 수 있다는 것이다. 근방은 위상 공간 속의 기본적인 개념의 하나로, 열린집합과 내부의 개념과도 밀접히 연관되어 있다.

정의

점의 근방

위상 공간 $X$ 속의 점 $x\in X$ 의 근방은 x를 열린 부분집합의 원소로 포함하는 집합이다. 즉, 어떤 열린 집합 $U$ 에 대하여 $X\supset V\supset U\ni x$ 가 성립할 경우, $V\subset X$ 를 x의 근방이라 한다.

점 $x\in X$ 의 열린 근방은 열린집합인 근방이다. 즉, 어떤 열린 집합 $U$ 가 $U\ni x$ 를 만족시킨다면, $U$ 는 $x$ 의 열린 근방을 이룬다.

x의 빠진 근방(영어: deleted neighborhood)은 $V\setminus \{x\}$ 꼴의 집합이다. 빠진 근방은 이름과 달리 근방이 아니다.

집합의 근방

위상 공간 $X$ 속의 부분공간 $Y\subset X$ 의 근방은 $Y$ 를 열린 부분집합의 부분집합으로 가지는 집합이다. 즉, 어떤 열린집합 $U$ 에 대하여 $X\supset V\supset U\supset Y$ 가 성립할 경우, $V\subset X$ 를 $Y$ 의 근방이라 한다.

예

구체적인 공간에서의 근방은 다음과 같이 정의할 수 있다.

실수선에서의 정의

$x$ 를 임의의 실수라 하자. 이 때, 반지름이 $r$ 인 $x$ 의 근방 $N(x;r)$ 은 다음과 같은 집합으로 정의된다.

N(x;r)=\{y\in \mathbb {R} :|y-x|<r\}

즉, $\mathbb {R}$ 에서의 근방은 개구간 $(x-r,x+r)$ 과 같다. 또, 여기서 $x$ 가 빠진 집합을 반지름이 $r$ 인 $x$ 의 빠진 근방(deleted neighborhood) $N'(x;r)$ 이라 하고 다음과 같이 정의한다.

N'(x;r)=N(x;r)\setminus \{x\}

유클리드 공간에서의 정의

$\mathbb {R} ^{n}$ 에서 반지름이 $r$ 인 $\mathbf {x}$ 의 근방 $N(x;r)$ 은 다음과 같은 집합으로 정의된다.

N(\mathbf {x} ;r)=\{\mathbf {y} \in \mathbb {R} ^{n}:||\mathbf {y} -\mathbf {x} ||<r\}

정의에서 보다시피, $n=2$ 일 때는 $\mathbf {x}$ 가 중심이고 반지름이 $r$ 이며 경계가 빠진 원판을 의미하고, $n=3$ 일 때는 $\mathbf {x}$ 가 중심이고 반지름이 $r$ 이며 경계가 빠진 구를 의미한다.

마찬가지로 반지름이 $r$ 인 $\mathbf {x}$ 의 빠진 근방 $N'(\mathbf {x} ;r)$ 은 다음과 같은 집합으로 정의된다.

N'(\mathbf {x} ;r)=N(\mathbf {x} ;r)\setminus \{\mathbf {x} \}

외부 링크

Pasynkov, B.A. (2001). “Neighbourhood”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. ISBN 978-1-55608-010-4.
Barile, Margherita; Weisstein, Eric W. “Neighborhood”. 《Wolfram MathWorld》 (영어). Wolfram Research. CS1 관리 - 여러 이름 (링크)

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